Comparison of symbolic and numerical integration methods for an assumed‐stress hybrid shell element

Revista de la Facultad de Ingeniería

et al. 2016

ResumenEn el ejercicio de la ingeniería, el análisis matricial de medios continuos tiene hoy una aplicación más generalizada, como consecuencia del desarrollo computacional, optimizando así los métodos y técnicas utilizadas mediante los Sistemas de Álgebra Computacional (SAC), estos permiten obtener expresiones analíticas en diversas aplicaciones en la Ingeniería así como académicas y en la industria.Este trabajo presenta la integración analítica aplicada a un elemento cuadrilátero " serendipity" de 8 nodos y de tipo subparamétrico. El propósito de este trabajo es calcular los términos de la matriz de rigidez para obtener mejoras en los tiempos de cálculo matemáticos en computadora además de aumentar la precisión en los cálculos de la matriz de rigidez para este tipo de elemento.Para lograr este objetivo se utilizaron programas que utilizan técnicas de matemática simbólica tales como el MAPLE, DERIVE entre otros, estos programas permitieron obtener 6 ecuaciones que calculan todos los términos de la matriz de Rigidez, estas expresiones se manipularon agrupando en constantes los factores de las expresiones de los términos de la matriz de Rigidez, quedando así expresiones algebraicas más sencillas en función de las variables naturales (ε, η). Estas constantes son a su vez función también de las coordenadas cartesianas de los nodos y de los parámetros elásticos del material, módulos de Young y Poisson, esto facilitó la integración analítica para cada término en la matriz, adicionalmente se crearon códigos que permiten calcular todos estos términos utilizando un lenguaje de alto nivel tipo Fortran.Palabras Claves: Manipulación simbólica en MEF, Integración exacta, Elementos finitos planos de 8 nodos, Matriz de rigidez, Deformación plana. ANALYTIC INTEGRATION OF THE STIFFNESS MATRIX OF A 8-NODED PLANE CUADRILATERAL SUBPARAMETRIC FINITE ELEMENT ABSTRACTComputer Algebra Systems (CAS) are powerful tools to obtain analytical expressions for many engineering applications in both academic and industrial environments.The explicit (analytical) for of the stiffness matrix of an 8-noded plane-elasticity finite element is obtained and described in this paper. Six basic formulas to generate the stiffness

Section: Antecedentesunclassified

Analytic Integration of the Stiffness Matrix of a 8-Noded Plane Cuadrilateral Subparametric Finite Element

Videla

Baloa

Revista de la Facultad de Ingeniería

et al. 2016

“…Zienkiewicz et al [6] present a semianalytical integration of plates and shells, and report a reduction in CPU time of nearly 50%. Rengarajan et al [7] integrated analytically a hybrid finite element for shell analysis and carried out comparisons between analytical and numerical integration. One of the advantages of hybrid finite elements is that the determinant of the Jacobian matrix does not appear in the denominator of the stiffness matrix expression as it does in conventional displacement-based finite element formulations.…”

Section: Introductionmentioning

confidence: 99%

Exact integration of the stiffness matrix of an 8‐node plane elastic finite element by symbolic computation

Videla

Baloa

Numerical Methods Partial

et al. 2007

Computer algebra systems (CAS) are powerful tools for obtaining analytical expressions for many engineering applications in both academic and industrial environments.CAS have been used in this paper to generate exact expressions for the stiffness matrix of an 8-node plane elastic finite element. The Maple software system was used to identify six basic formulas from which all the terms of the stiffness matrix could be obtained. The formulas are functions of the Cartesian coordinates of the corner nodes of the element, and elastic parameters Young's modulus and Poisson's ratio.Many algebraic manipulations were performed on the formulas to optimize their efficiency. The redaction in CPU time using the exact expressions as opposed to the classical Gauss-Legendre numerical integration approach was over 50%. In an additional study of accuracy, it was shown that the numerical approach could lead to quite significant errors as compared with the exact approach, especially as element distortion was increased.

“…Zienkiewicz et al [7] present a semi-analytical integration of plates and shells, and report a reduction in CPU time of nearly 50%. Rengarajan et al [8] analytically integrated a hybrid finite element for shell analysis and carried out comparisons between analytical and numerical integration. One of the advantages of hybrid finite elements is that the determinant of the Jacobian matrix does not appear in the denominator of the stiffness matrix expression as it does in conventional displacement based finite element formulations.…”

mentioning

confidence: 99%

Semi‐analytical integration of the 8‐node plane element stiffness matrix using symbolic computation

Lozada

Osorio

Numerical Methods Partial

et al. 2005

The semi-analytical integration of an 8-node plane strain finite element stiffness matrix is presented in this work. The element is assumed to be super-parametric, having straight sides. Before carrying out the integration, the integral expressions are classified into several groups, thus avoiding duplication of calculations. Symbolic manipulation and integration is used to obtain the basic formulae to evaluate the stiffness matrix. Then, the resulting expressions are postprocessed, optimized, and simplified in order to reduce the computation time. Maple symbolic-manipulation software was used to generate the closed expressions and to develop the corresponding Fortran code. Comparisons between semi-analytical integration and numerical integration were made. It was demonstrated that semi-analytical integration required less CPU time than conventional numerical integration (using Gaussian-Legendre quadrature) to obtain the stiffness matrix.